FOM: The meaning of truth

[JoeShipman@aol.com]

Professor Kanovei, I will boil down what I am trying to say to two points, 
since you keep focusing on what I had regarded as peripheral aspects of my 
postings (my fault for not reemphasizing the saliency of the following):

1) If you accept proofs from ZFC as valid, then why is the standard 
definition of truth for arithmetical sentences, as commonly formalized in 
ZFC, insufficient?  Why are you willing to accept as validly proved a theorem 
such as the Paris-Harrington theorem, or the finite form of Kruskal's 
theorem, which depend essentially on the axiom of infinity, but not willing 
to accept a definition of arithmetical truth which can be developed 
straightforwardly from the same axiom?

2) My comments and speculations about mathematical sentences which could be 
"scientifically proven" but not "mathematically proven" do not contradict 
anything you said, because I am willing to grant that the notion of "truth" I 
attribute to such sentences is inferior to the kind of "truth" you are able 
to attribute to sentences that have been mathematically proven.  But your 
insistence that existence of a mathematical proof of A is the ONLY way we can 
interpret "A is true" seems to me to unjustifiably dismiss the correspondence 
between physics and mathematics that has been the central theme of our 
science for centuries.  If you deny that physical theories (and the machines 
we can build based on them) can be sufficiently well-understood that 
experimental results can be interpreted as telling us something about 
mathematics, then it is hard to see why you accept the 4-color theorem as 
having the same status as the prime number theorem or other theorems that 
have been humanly verified.

-- Joe Shipman